Summary form only given. We define the voltage across an area of a resistive network by suitably combining voltages at buses (nodes) on the border of the area. The new voltage across the area satisfies circuit laws. The same concept works to define the voltage angle difference across an area of a DC load flow network and the complex voltage difference across an area of an AC load flow network. We first define the voltage across a cutset of lines, and then derive and explain the voltages across areas, including generalizations to several voltages across an area and multiple areas. The new voltages across areas seem promising for both power system monitoring and network reduction, and we describe their application to monitoring area stress.